Sendai City Museum is the best place to ground a visit to the old castle area in real history. Its collections focus on the Date clan, Sendai's founding, samurai culture, and the city's links beyond Japan, including material connected with the Keicho embassy to Europe.
The museum sits near the Aoba Castle site, so the two sights belong together. See the museum first if you want the ruined castle terraces and city views to mean more than a scenic overlook.